Short Answer 
The probability of all three coins landing heads is 1/8, while the probability of the sum of two dice being 10 is 1/12. Since these events are independent, the combined probability of either event occurring is 5/24.
Step 1: Calculate the Probability of All Coins Landing Heads
To find the probability of all three coins landing heads, we need to consider the total outcomes. Since each coin has two sides (heads or tails), the total number of outcomes for three coins is 2^3 = 8. Only one of these outcomes is the desired event of all heads, so the probability is calculated as follows:
- Total Outcomes: 8
- Favorable Outcomes: 1 (HHH)
- Probability: 1/8
Step 2: Calculate the Probability of the Sum of the Dice Being 10
For the sum of two dice to be 10, we first identify the favorable pairs that achieve this sum. The pairs that add up to 10 are (4,6), (5,5), and (6,4), giving us three favorable outcomes out of the total possible outcomes of 36 when rolling two dice:
- Favorable Outcomes: 3 (4,6), (5,5), (6,4)
- Total Outcomes: 36
- Probability: 3/36 = 1/12
Step 3: Combine the Probabilities for Either Event
To find the overall probability of either all coins being heads or the sum of the dice being 10, you simply add the individual probabilities obtained in the previous steps. It’s important to note that these two events are independent, so we don’t need to worry about any overlap between them:
- Probability of All Heads: 1/8
- Probability of Sum of Dice as 10: 1/12
- Combined Probability: 1/8 + 1/12 = 5/24